13.1 – The Curvature

We now know for a fact that the Delta of an option is a variable, as it constantly changes its value relative to the change in the underlying. Let me repost the graph of the delta’s movement here –

If you look at the blue line representing the delta of a call option, it is quite clear that it traverses between 0 and 1 or maybe from 1 to 0 as the situation would demand. Similar observations can be made on the red line representing the put option’s delta (except the value changes between 0 to -1). This graph reemphasizes what we already know i.e the delta is a variable and it changes all the time. Given this, the question that one needs to answer is –

I know the delta changes, but why should I care about it?

If the change in delta really matters, how do I estimate the likely change in delta?

We will talk about the 2nd question first as I’m reasonably certain the answer to the first question will reveal itself as we progress through this chapter.

As introduced in the previous chapter, ‘The Gamma’ (2nd order derivative of premium) also referred to as the curvature of the option gives the rate at which the option’s delta changes as the underlying changes. The gamma is usually expressed in deltas gained or lost per one point change in the underlying – with the delta increasing by the amount of the gamma when the underlying rises and falling by the amount of the gamma when the underlying falls.

For example consider this –

Nifty Spot = 8326

Strike = 8400

Option type = CE

Moneyness of Option = Slightly OTM

Premium = Rs.26/-

Delta = 0.3

Gamma = 0.0025

Change in Spot = 70 points

New Spot price = 8326 + 70 = 8396

New Premium =??

New Delta =??

New moneyness =??

Let’s figure this out –

Change in Premium = Delta * change in spot i.e 0.3 * 70 = 21

New premium = 21 + 26 = 47

Rate of change of delta = 0.0025 units for every 1 point change in underlying

Change in delta = Gamma * Change in underlying i.e 0.0025*70 = 0.175

New Delta = Old Delta + Change in Delta i.e 0.3 + 0.175 = 0.475

New Moneyness = ATM

When Nifty moves from 8326 to 8396, the 8400 CE premium changed from Rs.26 to Rs.47, and along with this the Delta changed from 0.3 to 0.475.

Notice with the change of 70 points, the option transitions from slightly OTM to ATM option. Which means the option’s delta has to change from 0.3 to somewhere close to 0.5. This is exactly what’s happening here.

Further let us assume Nifty moves up another 70 points from 8396; let us see what happens with the 8400 CE option –

Old spot = 8396

New spot value = 8396 + 70 = 8466

Old Premium = 47

Old Delta = 0.475

Change in Premium = 0.475 * 70 = 33.25

New Premium = 47 + 33.25 = 80.25

New moneyness = ITM (hence delta should be higher than 0.5)

Change in delta =0.0025 * 70 = 0.175

New Delta = 0.475 + 0.175 = 0.65

Let’s take this forward a little further, now assume Nifty falls by 50 points, let us see what happens with the 8400 CE option –

Old spot = 8466

New spot value = 8466 – 50 = 8416

Old Premium = 80.25

Old Delta = 0.65

Change in Premium = 0.65 *(50) = – 32.5

New Premium = 80.25 – 32. 5 = 47.75

New moneyness = slightly ITM (hence delta should be higher than 0.5)

Change in delta = 0.0025 * (50) = – 0.125

New Delta = 0.65 – 0.125 = 0.525

Notice how well the delta transitions and adheres to the delta value rules we discussed in the earlier chapters. Also, you may wonder why the Gamma value is kept constant in the above examples. Well, in reality the Gamma also changes with the change in the underlying. This change in Gamma due to changes in underlying is captured by 3rd derivative of underlying called “Speed” or “Gamma of Gamma” or “DgammaDspot”. For all practical purposes, it is not necessary to get into the discussion of Speed, unless you are mathematically inclined or you work for an Investment Bank where the trading book risk can run into several $ Millions.

Unlike the delta, the Gamma is always a positive number for both Call and Put Option. Therefore when a trader is long options (both Calls and Puts) the trader is considered ‘Long Gamma’ and when he is short options (both calls and puts) he is considered ‘Short Gamma’.

For example consider this – The Gamma of an ATM Put option is 0.004, if the underlying moves 10 points, what do you think the new delta is?

Before you proceed I would suggest you spend few minutes to think about the solution for the above.

Here is the solution – Since we are talking about an ATM Put option, the Delta must be around – 0.5. Remember Put options have a –ve Delta. Gamma as you notice is a positive number i.e +0.004. The underlying moves by 10 points without specifying the direction, so let us figure out what happens in both cases.

Now, here is trick question for you – In the earlier chapters, we had discussed that the Delta of the Futures contract in always 1, so what do you think the gamma of the Futures contract is? Please leave your answers in the comment box below :).

13.2 – Estimating Risk using Gamma

I know there are many traders who define their risk limits while trading. Here is what I mean by a risk limit – for example the trader may have a capital of Rs.300,000/- in his trading account. Margin required for each Nifty Futures is approximately Rs.16,500/-. Do note you can use Zerodha’s SPAN calculator to figure out the margin required for any F&O contract. So considering the margin and the M2M margin required, the trader may decide at any point he may not want to exceed holding more than 5 Nifty Futures contracts, thus defining his risk limits, this seems fair enough and works really well while trading futures.

But does the same logic work while trading options? Let’s figure out if it is the right way to think about risk while trading options.

Here is a situation –

Number of lots traded = 10 lots (Note – 10 lots of ATM contracts with delta of 0.5 each is equivalent to 5 Futures contract)

Option = 8400 CE

Spot = 8405

Delta = 0.5

Gamma = 0.005

Position = Short

The trader is short 10 lots of Nifty 8400 Call Option; this means the trader is within his risk boundary. Recall the discussion we had in the Delta chapter about adding up the delta. We can essentially add up the deltas to get the overall delta of the position. Also each delta of 1 represents 1 lot of the underlying. So we will keep this in perspective and we can figure out the overall position’s delta.

Delta = 0.5

Number of lots = 10

Position Delta = 10 * 0.5 = 5

So from the overall delta perspective the trader is within his risk boundary of trading not more than 5 Futures lots. Also, do note since the trader is short options, he is essentially short gamma.

The position’s delta of 5 indicates that the trader’s position will move 5 points for every 1 point movement in the underlying.

Now, assume Nifty moves 70 points against him and the trader continues to hold his position, hoping for a recovery. The trader is obviously under the impression that he is holding 10 lots of options which is within his risk appetite…

Let’s do some forensics to figure out behind the scenes changes –

Delta = 0.5

Gamma = 0.005

Change in underlying = 70 points

Change in Delta = Gamma * change in underlying = 0.005 * 70 = 0.35

New Delta = 0.5 + 0.35 = 0.85

New Position Delta = 0.85*10 = 8.5

Do you see the problem here? Although the trader has defined his risk limit of 5 lots, thanks to a high Gamma value, he has overshot his risk limit and now holds positions equivalent to 8.5 lots, way beyond his perceived risk limit. An inexperienced trader can be caught unaware of this and still be under the impression that he is well under his risk radar. But in reality his risk exposure is getting higher.

Now since the delta is 8.5, his overall position is expected to move 8.5 points for every 1 point change in the underlying. For a moment assume the trader is long on the call option instead of being short – obviously he would enjoy the situation here as the market is moving in his favor. Besides the favorable movement in the market, his positions is getting ‘Longer’ since the ‘long gamma’ tends to add up the deltas, and therefore the delta tends to get bigger, which means the rate of change on premium with respect to change in underlying is faster.

Suggest you read that again in small bits if you found it confusing.

But since the trader is short, he is essentially short gamma…this means when the position moves against him (as in the market moves up while he is short) the deltas add up (thanks to gamma) and therefore at every stage of market increase, the delta and gamma gang up against the short option trader, making his position riskier way beyond what the plain eyes can see. Perhaps this is the reason why they say – shorting options carry huge amount of risk. In fact you can be more precise and say “shorting options carries the risk of being short gamma”.

Note – By no means I’m suggesting that you should not short options. In fact a successful trader employs both short and long positions as the situation demands. I’m only suggesting that when you short options you need to be aware of the Greeks and what they can do to your positions.

Also, I’d strongly suggest you avoid shorting option contracts which has a large Gamma.

This leads us to another interesting topic – what is considered as ‘large gamma’.

13.3 – Gamma movement

Earlier in the chapter we briefly discussed that the Gamma changes with respect to change in the underlying. This change in Gamma is captured by the 3rd order derivative called ‘Speed’. I won’t get into discussing ‘Speed’ for reasons stated earlier. However we need to know the behavior of Gamma movement so that we can avoid initiating trades with high Gamma. Of course there are other advantages of knowing the behavior of Gamma, we will talk about this at a later stage in this module. But for now we will look into how the Gamma behaves with respect to changes in the underlying.

Have a look at the chart below,

The chart above has 3 different CE strike prices – 80, 100, and 120 and their respective Gamma movement. For example the blue line represents the Gamma of the 80 CE strike price. I would suggest you look at each graph individually to avoid confusion. In fact for sake of simplicity I will only talk about the 80 CE strike option, represented by the blue line.

Let us assume the spot price is at 80, thus making the 80 strike ATM. Keeping this in perspective we can observe the following from the above chart –

Since the strike under consideration is 80 CE, the option attains ATM status when the spot price equals 80

Notice the gamma value is low for OTM Options (80 and above). This explains why the premium for OTM options don’t change much in terms of absolute point terms, however in % terms the change is higher. For example – the premium of an OTM option can change from Rs.2 to Rs.2.5, while absolute change in is just 50 paisa, the % change is 25%.

The gamma peaks when the option hits ATM status. This implies that the rate of change of delta is highest when the option is ATM. In other words, ATM options are most sensitive to the changes in the underlying

The gamma value is also low for ITM options (80 and below). Hence for a certain change in the underlying, the rate of change of delta for an ITM option is much lesser compared to ATM option. However do remember the ITM option inherently has a high delta. So while ITM delta reacts slowly to the change in underlying (due to low gamma) the change in premium is high (due to high base value of delta).

You can observe similar Gamma behavior for other strikes i.e 100, and 120. In fact the reason to show different strikes is to showcase the fact that the gamma behaves in the same way for all options strikes

Just in case you found the above discussion bit overwhelming, here are 3 simple points that you can take home –

Delta changes rapidly for ATM option

Delta changes slowly for OTM and ITM options

Never short ATM or ITM option with a hope that they will expire worthless upon expiry

OTM options are great candidates for short trades assuming you intend to hold these short trades upto expiry wherein you expect the option to expire worthless

13.4 – Quick note on Greek interactions

One of the keys to successful options trading is to understand how the individual option Greeks behave under various circumstances. Now besides understanding the individual Greek behavior, one also needs to understand how these individual option Greeks react with each other.

So far we have considered only the premium change with respect to the changes in the spot price. We have not yet discussed time and volatility. Think about the markets and the real time changes that happen. Everything changes – time, volatility, and the underlying price. So an option trader should be in a position to understand these changes and its overall impact on the option premium.

You will fully appreciate this only when you understand the cross interactions of the option Greeks. Typical Greek cross interactions would be – gamma versus time, gamma versus volatility, volatility vs time, time vs delta etc.

Finally all your understanding of the Greeks boils down to a few critical decision making factors such as –

For the given market circumstances which is the best strike to trade?

What is your expectation of the premium of that particular strike – would it increase or decrease? Hence would you be a buyer or a seller in that option?

If you plan to buy an option – is there a realistic chance for the premium to increase?

If you plan to short an option – is it really safe to do so? Are you able to see risk beyond what the naked eyes can spot?

The answers to all these questions will evolve once you fully understand individual Greeks and their cross interactions.

Given this, here is how this module will develop going further –

So far we have understood Delta and Gamma

Over the next few chapters we will understand Theta and Vega

When we introduce Vega (change in premium with respect to change in volatility) – we will digress slightly to understand volatility based stoploss

168 comments

hi, went through the chapter and found the presentation a little harried than the others. calculations were off the mark and in places no decimal points are incorporated. karthik, were you not involved in this chapter?

Gamma value (in fact delta, theta, and vega) values are all calculated using a mathematical model called “Black & Scholes Options Pricing Model”. We will briefly talk about this in the coming chapters.

Hi kartik.
Very very thanks for this chapter. Although chapter has some minute printing mistakes of decimals and some calculation mistake(which doesn’t matters at all) but it is a very good and clear chapter which boosts my concept about gamma.
I have two questions-
1. In the topic estimating risk using gamma you have given-
New position delta = 0.85*10=25. What is means??? Please clarify
2. Suppose , for any underlying–
Spot price=8000
Strike price=8200
Premium=100
Option type=CE
After sometime,
Spot price 9000
Premium=800
If one square off his position at current price, will his position be square off because deep itm options have very less volume. Is any volume issue occurs???
Futures have gamma=0, since delta is always 1(answer to your question)

I think I know today why they say that don’t trade options if you are a novice and blindly buy OTM options with the hope that you’d become rich overnight 🙂

I can appreciate complex mathematics Karthik – Would you suggest any book in particular (preferably by an Indian author who has kept indian indicies and stocks in perspective) for options? I am suddenly getting very interested and I will like to utilize the interval between which the chapters are posted here.

Couple of other generic questions:

1. Does it make sense to trade naked call/put options with say 1 lot nifty or is it just a bad idea altogether?
2. In your practical experience, are there folks who only trade options and nothing else?
3. In your practical experience, is it very difficult to make money with options? Also, what is better? Trading intraday or with 1/2/3 months time frame?
4. This one is really from the heart: While we know a lot of super rich value investors, why do we not hear of any super rich derivative traders? (Please do not take this question otherwise – Being an individual retail investors feels like a closed door with not much authentic information. Since you are hands-on in the field, you’d be the appropriate person to ask)

Saurabh – Options trading is a different animal altogether. One needs to be aware of many different aspects involved. I’m really not sure which book to recommend, I found some of the books that I’ve read previously quite shallow. Maybe I’ve not updated my library so may not be the best person to answer this for you.

Coming to your other questions –

1) No harm trading 1 or 2 lots of naked options – lot size is not much so you don’t risk anything significant. But neither will this create any significant wealth for you. In fact I do this quite often just to test few random ideas.

2) Yes there are many people trading just options – you can trust me on this

3) You certainly can make money trading options – but you need to set your expectations right (ref section 1.3 – http://zerodha.com/varsity/chapter/background/). Trust me if you can consistently make 2% a month trading options you are doing a phenomenal job. Unfortunately most people perceive this a small return and aim for a higher return, in the process blowing up their account. So my take on this – aim for small but CONSISTENT return. Also, I prefer to trade with slightly longer duration than intraday. I avoid intraday because I don’t find the time and commitment it requires.

4) You have a valid point – I agree we all get to read about successful Value Investors but not many derivatives traders. I really dont know why. But this does not mean they dont exist. There are many traders who generate significant amount of money trading derivatives…cant speak on behalf of other brokers but we do have some really good traders with us. We had the opportunity to interview some of them here – https://zerodha.com/z-connect/category/zerodha-60-day-challenge

Good luck and please stay tuned. I can guarantee you that Zerodha will put up everything you need to know about Options trading, just give us a little more time 🙂

Does zerodha provide a virtual trading a/c? It would be of great help if there is. I tried the one on NSE (Pathshala) but don’t get how it is showing my MTM as +ve when I in fact know that it is in the red. A simulated environment would really help!

Thanks for easy explanation on gamma. You had asked about gamma for the future contract. It is having fixed delta, 1 so no change in the delta means gamma of future contract will be zero. Please correct me if I am wrong.

I’ve personally used Options Oracle extensively, I must admit the software was one of its kind. I guess they stopped supporting Indian markets and hence I lost interest..or something else happend which stopped me from using OO. Cant recollect the exact reason.

First of all; excellent ,excellent work…..I have read a lot of material (books, websites, platforms, trading mentors etc etc) but none of them have been as concise and simple as your content is. You can really “Teach” ..hats off. ….
Coming to my question:
1) If i am a regular nifty futures trader, trading say 10 lots on a regular basis, i can instead trade 20 lots of nifty ATM options (only buy CE/PE) for the same effect (positive or negative return) …..If my trade is in my favor and delta keeps on increasing, i need not adjust my position since the increased delta is improving my return……However, if it is going against me , i may need to adjust (reduce) my position based on the delta of the CE/PE i am holding???? If i am only buying ATM options, position going against me may also mean that delta is reducing ? and therefore my risk may also be coming down? It is more a question of the rate of change…..i.e the reduced position delta is within my initial risk range or not???

I’m not sure if I’ve understood your question completely…but if it is what I think, then here are my thoughts –

If you buy an ATM option and the position starts to move in your favor the both Gamma and Delta works in your favor as the option transitions from ATM to ITM. However if you short options (ATM in this case) then more than the Delta you need to be worried about Gamma as you are essentially short Gamma when you short options …and ATM options have the highest Gamma value so you are taking on a large risk.

This also means you need to constantly tweak your position to ensure you are always short ATM option and not really ITM or OTM. This incurs logistical expenses…for that sense its better to just trade futures. However there are many trading secretaries (like Dynamic Delta hedging) which requires you to substitute options for futures. We will talk about it more in the subsequent modules.

Ok Thanks for your response…..For clarity, lets take the below scenario…..As of now, assume i am long 10 futures at 8471. I have my stop around 8400……..Now lets assume instead of futures, i have 20 option lots of 8450 CE…….When it goes in favor, it will probably give me more than futures….But if it goes against me, lets say Nifty closes at 8430 tomorrow, am i likely to lose more money than in options compared to what i would have lost in futures…Guess i am trying to understand what happens to delta when a position goes back into slightly OTM from ATM and then if the position needs to adjusted accordingly (in case of LOng CE @8450)

So as you can see once the option moves away from ATM status, it no longer becomes a surrogates for the futures contract. In order to maintain the equivalence you always need to adjust the option to ensure you are dealing with ATM options.

Is it possible to define gamma w.r.t. delta in place of underlying price as the second derivative may define based on the first derivative also apart from the base variable. Like distance, change in distance is velocity and change in velocity is acceleration. This may bring the delta calculation direct.
My second question is that why we have to worry more about gamma when options are at ATM or otherwise as gamma is constant for practical purpose………………..Why short position are more affected by gamma?

Yes, you can treat Gamma as the 1st derivative as Delta, no harm with that. When the option hits ATM status, then Gamma hits maximum value. See the graph posted in the chapter. Also since Gamma is always a positive number, shorting options carry the risk of “Short Gamma’. Hence I say short positions carry ‘short gamma’ risk. However this should discourage you for shorting options, juts bear in mind the fact that when you short Options, you are short gamma.

Good work Karthik. Really liked the way the concepts has been put across in a very simple language. Is it possible to provide a module to explain starting from TA on an underlying to identify the direction of the underlying and thereby identifying a strike price using greeks.

In extension of Amit post: Actually the spot movement depends on many factor like TA, fundamentals and current news, global developments etc. This is a normal concept. I want to know is it possible that TA will be always dominating other factors? How to judge the more effective factors. At times we have seen that even after the quarterly results are good the script price goes down and vice versa. Is it that there TA was having more power? Examples are SBI, Infy, Kotak B etc.

You always need to take a holistic view. There are many factors that affect the stock price form a short term perspective, although only business fundamentals matter in the long run. As a trader you need to develop a sense for all these factors and never isolate a certain factor as more dominant. How to judge these factors and assign a weight to the price movement is something you will develop as you spend more time in markets.

Thanks,
But options and future are valid for 1 month only or future may be for few months which I thinks is shot term and we have to analyse the market like a short term trader. Hence for Options all factors will matter. How to slowly rope in all factors may be of option’s trading. This was my point.

Understood your point, one of the readers (I guess Amit) suggested that I do a case study where I consider all points and factors and plug things together. I think thats a great idea I will do it towards the end of this module. That should answer most of your concerns.

The current focus is to finish Option Theory…few more chapters here…once we are done with Options Theory, we will start work on Options Strategies. I guess sometime next month we should get started on Options Strategies.

Hi Karthik,
Thanks for reply… and also I would like to thank you for such good quality, easy to understand study material by you on Varsity. I never seen such material on internet and too at one place. I am new to share market, I was doing trading blindly, now stopped everything and doing study with Varsity modules. Really thanks for your great efforts. Please keep it up…God Bless You…:)
Regards,
Abhijit

The gama decreases when a call option transits from ATM to OTM and ATM to ITM. due to the lower gama the change in premium is minimal with respect to the change in underlying. since, the ITM call options have a higher delta the change in premium in points term is higher even though the gama plays the drag. would this be a correct assessment karthik?

You said “The gama decreases when a call option transits from ATM to OTM and ATM to ITM. due to the lower gama the change in premium is minimal with respect to the change in underlying” —-> This is correct, however do remember in terms of absolute points change it maybe minimum but % wise, it will be a reasonable number.

“since, the ITM call options have a higher delta the change in premium in points term is higher even though the gama plays the drag” —–> Right on. However Delta is highest when Options are near ITM…but at ITM gamma is lower. ATM delta is around 0.5, but the sensitivity at this point is high as the gamma is the highest at ATM.

Gamma of futures contract should be “zero” as the delta of futures contract does not change at all so the rate of change is constant so the Gamma of futures should always be constant and because change in delta of futures contract because of change in spot price change is zero so the gamma of futures contract should be zero

There is a mistake from you in chapter number 13.3.
You wrote that when the spot price is below 80, the call option with strike 80 is in the money and above 80 is out of the money. It should be other way around. Spot above 80 should be in the money while below 80 should be out of the money.
PLEASE CORRECT THIS MISTAKE>

Awesome is the word for your efforts, knowledge and presentation skills Karthik, I am trading (or I must say gambling 🙂 ) options since 2007 with absolutely no knowledge of all these factors and now I know why am I in huge loss :). I am still unable to grasp all of it but this is the best thing so far and my dream of becoming full time trader now has a new hope !!!
Question :- When does the delta change for given option? for example if spot is at 8000 and 8050 ce is OTM with delat say 0.3 and spot moves 100 points to 8100 in 4 hours making 8050 ITM at what point will delta change? is it time dependent i.e end of market or start of market or is it price dependent i.e every 20 points etc?

Hi
When i go to http://www.moneycontrol.com/stocks/fno/marketstats/options/gainers/index.php
and filter for top gainers , all options , CE , all expiry . I seem to get companies with phenomenal growth for the day (eg 14900% in a day for Bajaj auto CE 2700 today )
but when i check that same option at that same strike price and same expiry i get completely different numbers in the option chain for Bajaj Auto.
Am i doing something wrong ?
These numbers are different every day across moneycontrol and NSE option chain .

thanks for your reply one more question/confusion, point 3 &5 below the greeks graph says

“Notice the gamma value is low for spot values between ATM and ITM (80 and above).” — point 3
“The gamma starts to decrease when the strike starts to transition from ATM to OTM (80 and below)” — point 5
Should it not be ITM (80 and BELOW) for point 3 and
OTM(80 and ABOVE) for point 5?

Sir, Today Heromotor spot price went up from 2714 to 2738 almost 24 rs up. but ce 2850 for July hardly changed from 4.6 to 4.65 (.05) only whereas ce 2850 for August rose from 33.0 to 45.0 significant rise. Why its so. Cause July ce will have less time value hence IV change shall be % wise more and August ce will have more time value so effect of IV shall be less % wise. Also, what about IV change both should be same or not? This is crucial from selecting not only strike price but also expiry date also. Please elaborate. –Thanks

The gamma for Futures (also other cash products) where delta is 1 meaning the rate of change in price is in perfect correlation to the change in Price of the underlying, would be Flat, since there is no curvature on the Delta curve during the life of the product.

Both the statements hinges on the same fact – towards the expiry OTM and ITM options have a great chance to remain OTM or ITM, hence they can expire worthless….therefore they are great candidates for shorting. Whereas ATM could be a bit tricky…it has a 50 – 50 percent chance of expiring as a worthy contract/not worthy contract…hence no point taking the risk to short these options.

After getting inspired I am planing to write Call options that are slightly OTM.

Could you please tell when exactly an option is considered to be ATM. Is it when strike Price falls within some price band around spot price say when Strike Price = Spot Price ± some %age of Spot price ? What is that %age ?

What ideally should be delta, gamma and theta values to write OTM call options ?

Hi sir
I am really thankful to you for providing so many insightful articles. Its very easy to read and best part is its applicability in the real scenario. You are amazing.
I have one query. In 13.2 example … since we are short on 10 lots Nifty 8500CE so Delta would be -ve as i have done below. Please clarify.
Number of lots = 10
Position Delta = -10 * 0.5 = – 5

1) first we consider 70 points upwards , premium appreciates .
instead of considering a 50 point downward movement . let us consider a 70 point drop .
in this case the underlying is back to original state with no profit & no loss .
but when we calculate new premium , it is a t a loss and depreciates to 36.85 ….. am i correct or have i made a mistake ?

2) CASE-2 :-
the underlying falls 70 points and gains back 70 points…. again no profit ,no loss
but in this case also , the premium is at a loss at the value = 34.75 .

PLEASE DO CORRECT ME IF I AM WRONG .

in this kind of scenario , why & when should anyone consider option trading other than spot or future markets ?

You need to factor in the fact that with the change in underlying both the delta and Gamma changes. The calculations in the next step should be considered with the new values in perspective. Hope I’m not confusing you more 🙂

dear karthick,
i have a logical doubt . suppose the nifty spot is trading at 7800 , and i decide to buy 7600 call option it is deep itm contract. BUT IF IAM WRITING THE SAME 7600 CALL IT WOULD BE AN OTM CONTRACT ISNT IT. THE CONTRACT WHICH IS ITM FOR AN OPTION BUYER WOULD BE OTM FOR AN OPTION WRITER ISNT IT

Dear Karthik Sir Thanks for yet another good article on options on scientific background. You triggered my memories on Differential Calculus and Physics. The delta vs spot_price curve looks to have a perfect mathematical pattern. I found the curve y = 1 / (1 + (1/(x^4))) is best matching the delta vs spot_price curve. Let O be Deep OTM, A be ATM, I be deep ITM and S be spot price. Then x = ((O – S)(A -I)) / ((O – A)(S – I)). If you plot this, you can see the curve created matching delta curve. So gamma is first derivative of Y = 1 / (1 + (1/(x^4))) which is Gamma = (4(x^3)) /( ((x^4) + 1)^2)

Yes Gamma is the 1st order derivative…in fact you can differentiate this further to get the 2nd order derivative :). 2nd order is used to hedge posistions especially when you are running a huge trading book. For small retail positions this is not really required.

Above you mentioned ” Notice the gamma value is low for OTM Options (80 and above). This explains why the premium for OTM options don’t change much in terms of absolute point terms, however in % terms the change is higher.”, I get that premium doesn’t change much in terms of absolute points because of low delta but how does low gamma affect relative change in premium?

Well, if the gamma is low, then the premiums don’t change much. However if the volatility increases drastically or if the market moves in favor then option premiums for low gamma options tend to move…this is when you will see OTM options doubling in value. Recent example was the way L&T options reacted to a 14% increase in the stock price.

Hey Karthik,
Your every response to public doubts and the articles encourage people to trade before they opt out due to risk taking in markets.
I got confused with the below points, hope you’ll rectify
1. In your 1st two example showing a change in the delta with respect to Gamma with 70 points up (in each) changes the moneyness from OTM to ATM and lastly to ITM giving a premium of 80.25 keeping the New Delta at 0.65.
My question if Nifty changes by 140 points up directly (irrespective of 70, 70) although the moneyness becomes ITM from OTM. But giving a premium of 68 at Delta 0.65.
How to get a proper view on this ?
2. Every Underlying or Index has it’s lot size stated on NSE website. Can we choose lots below the stated. Say Nifty has 75 market lot, can we take 20 lots for trade or we have to trade on 75.

My bad, I completely mistook your question. There are two explanations for this –

1) Mathematical explanation – Technically speaking the change in premium, delta, and gamma are all defined for 1 point change in the spot price. So when I say there is a 70 point change in spot, I tend to directly multiply the 70 points change with delta (or gamma) to get the premium value..but note this is just an approximation. To get the true value for a 70 point change in spot, I will have to calculate the change in premium by changing the value of delta (and gamma), for 1 point at a time….and I will have to iterate this 70 times. This is when when i get true value of the premium. In fact gamma/delta are instantaneous values and I will have to do this one step at a time. An alternate way to do this is would be by integrating (recall your school calculus) over the change in premium.

2) Easy explanation – Note, when the spot price changed 70 for the 1st time, the delta changed from 0.3 to 0.47 and for the 2nd 70 points, the change in premium occurred on a higher delta, therefore higher premium. Whereas when you took the 140 point change directly, this happened on a smaller delta i.e 0.3, thus leading to a lesser premium when compared to two 70 point change

Calculus? You are kidding right!
But thanks, somewhat you have put me in one step further to dealt with my confusion although I still have to ask you for some more recognition
Say at 10 am = Bought 8400 CE @ 26 (Slightly OTM), spot 8326
11 am = Spot goes to 8396 @ 47 (ATM),
12 pm = Spot goes to 8466 @ 80.25 (ITM)
or else,
at 10 am = Bought 8400 CE @ 26 (Slightly OTM), spot 8326
12 pm = Spot goes to 8466 (140+) @ 68 (ITM)
So if I want to square off my position at 12 pm what will be the consequence according to the above follow up?

I have not disrespected your ways, it’s me poor in calculus. :'(
It means if price movement rests its value changes but during the movement, no value get affected? And if the movement lasts for less time the change in premium is high? Am I wrong somewhere?

1. When spot price starts to change, the other values of an option get affected once the spot price reached a higher/lower value than earlier.
2. The values don’t change until the spot price reach a predetermined price (either high or low)
3. If the spot price shows a big movement within few moments then the premium changes rapidly or price movements in large periods gives lower rate of premium change
Summary: Spot price 8326 changes 70 points twice within less time than the change of 140 points.
So, Big movements within less time , Higher premium
Big movements in higher time, Less premium
Hope you have understand my confusion.
Looking for your reply

In “Case 1 – Underlying moves up by 10 points”, the new delta should be -0.5 – 0.04 = -0.54 as delta is loses its value when the underlying price goes up so its value must decrease. It is also clear from graph of delta vs spot price for put options that value will decrease. Similarly in case 2 answer should be -0.46.

Delta of a Put option decreased when the underlying increase, hence-0.5+0.04 = -0.46 is correct. Remember, for Put option, delta is negative to capture the effect of decrease in delta for increase in underlying.

Right now I am holding 12900 qty of option 9500CE at the average price of rs.3.00. But currently the price is rs.1.15. Can you suggest what can I do now.can I hold it or sell it. Another 7 trading days only available for match. The loss is 23 k now.

On Sec 13.2 Estimating Risk Using Gamma
Position – Short
When the underlyning moved by 70 points would the position not slightly become OTM for him and delta of the over all position be reduced rather going up from 0.5 to 0.15 rather than 0.85

In the example of calculating risk using Gamma, shouldn’t we consider negative sign before the number of lots because we have taken a short position?
i.e. position delta = -10 * 0.5 = -5.
then, when the market moves against me, new gamma = 0.05*70 = 3.5.
therefore, shouldn’t the new delta be = -5+3.5 = -2.5?
Please correct me wherever I’m wrong.

i have a question which may sound funny. i have gone through all chapters till now but there are too many mathematical calculation & seriously looks horrible to me (not a academic person at all). my question is that-
“suppose i am very bearish on a particular contract and i decides to buy a way far out of money put with next month expiry (since i am extremely bearish on it) which is available at almost free of cost lets say may be around Rs. 0.80 – 1. so before initiating the trade should i look for delta & gamma, when there is nothing on stake.

As we have seen position of ATM options is most sensitive to changes in underlying therefore the effects of gamma on it but why is Gamma of an ATM option highest?
Also I wanted to know why theta of a deep put option is positive? (Though it should have been posted in other chapter, I am posting it here for the sake of convenience.)

This is an excellent series. I found it only in Sep 2017 and I am reading it for the ‘n’th time, just to get my fundas clear. I really appreciate your patience in answering all the questions too ! Hats off to you ! (I have just opened an account in Zerodha 🙂 )

Considering that this article was written in July 2015, I hope it is not too late for me to ask these questions:

1. I don’t think I have understood the difference between short gamma and long gamma. Is it that if you have taken a long position, it is long gamma and if you have taken a short position, it is short gamma ?

2. Let us say that an option is ATM, where the gamma is the highest. Now whether the price moves up or down, the gamma is going to decrease in the same way (because the gamma curve is like a bell curve – I have assumed it to be symmetrical about the ATM). With this, if I have shorted the option, the risk of price going up is the same as the reward of price going down, isn’t it ? So why is it dangerous to short an ATM option, considering that the reward is equally good ?

One reason I can think of is that as the price increases, the Delta also increases and hence it has a cascading effect (thus increasing my losses, if I have shorted) whereas when the price falls, Delta decreases so as to minimise the overall effect a bit, thus reducing my profits a bit. In other words, the losses I incur by a move of x points upwards in the spot price will be more than the profits I make by a move of x points downward. Is it ?

1) Yes, from the gamma perspective, long position is often called long gamma and short positions as short gamma
2) Since ATM options have the highest gamma they also tend to be the most sensitive options. It’s like this – assume you are short simultaneously on an ATM option and also short on an OTM option. Now if the market moves against you, then obviously you will lose money on both these short positions…but you will lose much more on the short ATM position compared to the OTM option.

I understand that the unfavourable market movement affects the ATM options more than OTM options because of high gamma of ATM. For example, if I have bought an OTM call and an ATM call, if the price comes down, the drop in premium is more for the ATM call, right ?

So let us say I have bought an OTM call and after a day, market moves favourably and now my option is ATM. I expect the price to go up further. Is it better to:

1. Leave it as it is (and risk the premium coming down steeply if the price comes down)
2. Square off the ATM call and buy an OTM call (I can even book the profit and but an OTM call with the original amount)

Yes, the ATM call drops more as the ATM option is most sensitive to directional movements i.e Delta
1) Depends on your trade set up and the targets you have set. Also, if you are holding an option which has transitioned from OTM to ATM, then you will have enough cushion in terms of profits.
2) Again, depends on your risk appetite 🙂

In the options calculator, what values should be taken as implied volatility and Interest? On the specific stock derivative on nse webpage their is a dash in place of value for volatility. Please enlighten. 🙂

Pretty sure your first diagram delta vs stock price is wrong for the put delta.
Put delta doesnt start at 0 and move to -1 it starts at -1 and moves to 0.
The two curves should be shifts of each other not mirror images.

First of all, thank you very much for putting up varsity. I learned a little bit of FA and TA through some classes. Futures and options classes are priced too costly and was searching for alternatives. That is when I landed here. When I finished all your options module, to my surprise, I noticed that I did not even take a single line of notes for the first time in learning anything in my life. All just went, straight into the head ;). Such was the manner how the concepts are explained. Referred many of my friends to this site.

I will be doing my first options trade next week :). So I am doing a quick recap of all the modules. While doing so, I also noticed that there is a slight confusion in the delta vs spot price figure. May be you need to remove the spot price caption at the bottom. Going by delta vs spot price in the figure, it conveys that as the spot price increases delta of the put option goes from 0 to -1. I think this is what Fbish pointed out. In such case for put option, the call option delta curve would just shift down.

If we want to go by OTM, ATM and ITM labels, then we may need to remove spot price caption below. It may lead one to interpret that spot price increases from left to right.

Thanks again Karthik Rangappa and Team Zerodha. All your sincere efforts are highly appreciated, deeply regarded and will be remembered through out my journey in the markets.

Vetriselvan, thank you so much for such kind words. I’m really happy you liked the content here.

Quite a few people have pointed this on the spot vs Delta chart. The trick is to start from the center of the graph and then look to the right (where the spot increases) and then to the left (where the spot decreases)…and observe the behavior of both Call and Put Delta. Anyway, I will see what I can do to make it easier.

Anyway, good luck for your very first options trade. Hope you learn and earn well!

You may consider putting separate labels for put and calls along the curves. Starting from the left for calls, it would be OTM, ATM and ITM and for puts it would be ITM, ATM and OTM with blue and red colors. Alternatively, you may just remove the labels and draw arrows from the centers towards the left and right saying decrease and increase in price respectively.

Thank you.Your material is simple to understand.
Please clarify my doubt
Initially, you are assigning values for delta as 0.2, 0.8 etc.. for Delta 0.05, 0.025 etc..
How can we fix those values? Are those are estimated values?
kindly reply me
Thank you

Hi
Here we are calculating delta change considering 70 points change in spot price.
for eg current delta is .3 and if gamma is .002 so new delta will be =.3 +(.002*70) equals to 0.44

Now I am sure in real world delta calculation happens for every single point change in spot price so for every point change in spot we use to get new delta as per calculation explained above. Am i right?

Great work Karthik!
I had a concern though, the statement ‘when you short options, you are shorting gamma’ is still unclear. If you could further delve into this aspect and enlighten us.
Thank you for the amazing write-up.

This is quite straightforward, Ashish. Gamma is always a positive number – and not like the DELTA which can take -ve value based on the position you take. So when you go long on an option, you are also long on the gamma, likewise, when you are short on an option, you are also short gamma.